Pulse Amplitude Modulation (PAM)#
PAM Waveform#
Pulse Amplitude Modulation (PAM) is a modulation scheme in which the amplitude of a pulse is varied according to the digital data being transmitted.
In digital PAM, the transmitted signal waveforms can be expressed as:
where:
\( p(t) \) represents a bandpass, real-valued pulse waveform with a duration \( T \).
\( \{A_m, 1 \leq m \leq M\} \) is a set of \( M \) possible amplitudes, which correspond to \( M \) distinct symbols (or constellation points) in the modulation scheme, and each of these symbols corresponds to a unique \( k \)-bit binary sequence.
Specifically:
\( M \) represents the number of distinct amplitude levels (i.e., symbols) in the Pulse Amplitude Modulation (PAM) scheme.
Since digital communication works with bits, we need to map binary data to these amplitude levels.
If each symbol represents a fixed number of bits, say \( k \), then the total number of distinct symbols must be \( 2^k \), since there are \( 2^k \) possible sequences of \( k \) bits.
This means that each PAM symbol (amplitude level) corresponds to one of \( k \)-bit binary sequences.
Example
Suppose 4-PAM is used (\( M = 4 \)). Since \( 4 = 2^2 \), each symbol represents 2 bits.
The four amplitude levels might be assigned as:
\( A_1 = -3 \) → “00”
\( A_2 = -1 \) → “01”
\( A_3 = +1 \) → “10”
\( A_4 = +3 \) → “11”
Similarly, for 8-PAM (\( M = 8 \)), since \( 8 = 2^3 \), each symbol represents 3 bits, and we would have 8 distinct amplitude levels.
Discrete Amplitude Levels in PAM#
The amplitude levels \( A_m \) in a typical PAM system take discrete values, commonly represented as:
This formulation results in symmetric amplitude levels around zero, meaning the signal amplitudes are distributed as:
These discrete levels enable efficient mapping of binary data onto amplitude variations.
Waveform Shape and Spectral Influence#
The waveform \( p(t) \) is also termed as a real-valued pulse signal, and its shape significantly impacts the spectral characteristics of the transmitted PAM signal. The choice of \( p(t) \) determines the bandwidth and spectral efficiency of the transmission. Common choices for \( p(t) \) include:
Rectangular pulses: Simple but spectrally inefficient.
Raised cosine pulses: Used to control intersymbol interference (ISI).
Energy of the PAM Signal#
Energy of Signal \( s_m(t) \)#
The energy associated with a transmitted PAM signal \( s_m(t) \) is given by:
Since the pulse \( p(t) \) is common for all \( s_m(t) \), its energy is given as:
Thus, the energy of the signal \( s_m(t) \) becomes:
Average Energy of All Signals in the PAM Constellation#
For a PAM system with \( M \) levels, the average energy is the mean of the energies of all \( M \) possible signals:
Substituting \( \mathcal{E}_m = A_m^2 \mathcal{E}_p \):
Summing the Amplitude Levels
The amplitude levels \( A_m \) are given by:
The square of these levels is:
Using symmetry in PAM levels (as shown earlier), this reduces to:
The series \( 1^2 + 3^2 + \dots + (M-1)^2 \) is a sum of squares of odd numbers. Using the formula for the sum of squares of the first \( n \) odd numbers:
Thus:
Substituting \( \sum_{m=1}^{M} A_m^2 \) into the average energy expression:
Simplify:
Normalize Average Energy at Bit-level#
For a PAM system with \( M \) levels, each symbol represents \( \log_2 M \) bits. The average energy per bit is:
Substituting \( \mathcal{E}_{\text{avg}} = \frac{(M^2 - 1) \mathcal{E}_p}{3} \):
Bandpass PAM and Equivalent Baseband (Lowpass)#
Pulse Amplitude Modulation (PAM) can be classified into baseband and bandpass forms based on what pulse shape is used.
If the pulse waveform \( p(t) \) is a bandpass signal, then we have bandpass PAM.
A lowpass equivalent representation of the PAM signal can be expressed as:
\[ \boxed{ s_{m,lp}(t) = A_m g(t) } \]where:
\( A_m \) represents the discrete amplitude levels of the PAM signal.
\( g(t) \) is a lowpass, real-valued pulse waveform that determines the spectral characteristics of the transmitted signal.
Mathematical Representation of Bandpass PAM#
For a bandpass signal, the transmitted PAM waveform is obtained by modulating the baseband equivalent onto a carrier frequency \( f_c \). This is expressed as:
Since the lowpass equivalent is given by \( A_m g(t) \), we can rewrite:
Expanding the real part:
where:
\( f_c \) is the carrier frequency, determining the central frequency of the transmitted signal.
\( g(t) \) is the baseband (lowpass) pulse shape.
Bandpass PAM and Amplitude-Shift Keying (ASK)#
If the transmitted PAM signal is of the generic form:
\[ \boxed{ p(t) = g(t) \cos(2\pi f_c t) } \]then the system is classified as bandpass PAM.
In digital communications, bandpass PAM is commonly referred to as Amplitude-Shift Keying (ASK). This is because the amplitude of the carrier is varied in discrete steps according to the digital data being transmitted.
Energy based on \(\mathcal{E}_g\) of \(g(t)\)#
We denote the pulse energy \( \mathcal{E}_g \).
Note that \(g(t)\) is lowpass signal, while \(p(t)\) is bandpass signal, we have
Thus, we have
Energy of a single signal:
Average energy across all signals
Average energy per bit
Representing PAM Signals Using Basis Functions#
Dimensionality of PAM Signals#
Pulse Amplitude Modulation (PAM) signals are fundamentally one-dimensional (\(N = 1\)), meaning that all possible signal waveforms can be represented as scaled versions of a single basic function.
Basis Function based on bandpass Pulse Shape \(p(t)\)#
Using the bandpass \(p(t)\), the basis function is defined as:
where:
\( p(t) \) is the pulse shape of the bandpass PAM signal.
\( \mathcal{E}_p \) is the energy of \(p(t)\).
\( \phi(t) \) is a normalized function, ensuring that the energy of \( \phi(t) \) is 1.
Using this basis function, any bandpass PAM signal can be represented as:
Basis Function based on Baseband Pulse Shape \(g(t)\)#
Using the baseband \(g(t)\), the basis function takes a different form due to the presence of a carrier frequency \( f_c \). The normalized basis function for bandpass PAM is:
where:
\( g(t) \) is the baseband pulse shape.
\( \mathcal{E}_g \) is the energy of \( g(t) \).
The factor \( \sqrt{2} \) ensures proper energy normalization in bandpass signaling, i.e., \(\mathcal{E}_p = \mathcal{E}_g/2\)
Using this basis function, the bandpass PAM signal is expressed as:
DISCUSSIONS
In both cases, \( \phi(t) \) is a bandpass basis function.
Specifically:
Case 1: Basis Function Using Bandpass \( p(t) \)
\( \phi(t) = \frac{p(t)}{\sqrt{\mathcal{E}_p}} \), where \( p(t) \) is the bandpass pulse \( g(t) \cos(2\pi f_c t) \).
Since \( p(t) \) is inherently a bandpass signal (due to the modulation by \( \cos(2\pi f_c t) \)), \( \phi(t) \) is also bandpass.
Case 2: Basis Function Using Baseband \( g(t) \)
The basis function in this case is explicitly modulated to a bandpass form: \(\phi(t) = \sqrt{\frac{2}{\mathcal{E}_g}} g(t) \cos(2\pi f_c t)\)
Although \( g(t) \) is a baseband (lowpass) signal, the inclusion of \( \cos(2\pi f_c t) \) shifts it to the bandpass spectrum.
Therefore, \( \phi(t) \) is still a bandpass function.
Why is \( \phi(t) \) Always Bandpass?
The goal of bandpass PAM is to represent signals in the bandpass spectrum, centered around the carrier frequency \( f_c \).
Energy Normalization in Both Cases
The normalization ensures that \( \phi(t) \) has unit energy (\( \int_{-\infty}^\infty \phi^2(t) dt = 1 \)) regardless of how it is constructed
Gray Coding and Binary Coding#
Mapping Information Bits to Signal Amplitudes#
In Pulse Amplitude Modulation (PAM), a set of \( k \) information bits must be mapped to one of \( M = 2^k \) possible signal amplitudes. The choice of how these bits are assigned to amplitude levels significantly impacts the system’s performance, particularly in the presence of noise.
Gray Coding#
Gray coding is a binary numeral system in which consecutive values differ by exactly one bit.
Example:
Convert decimal values \(0, 1, 2, 3, 4\) to Gray code (3 bits):
\(0 \to 000\)
\(1 \to 001\)
\(2 \to 011\)
\(3 \to 010\)
\(4 \to 110\)
1-bit difference example: Transition from \(2 \to 3\): \(011 \to 010\). Only the second bit changes.
Binary Coding#
Binary coding is a natural representation of numbers in the binary numeral system, where each number is represented by a unique binary sequence. Consecutive values may differ by multiple bits.
Example:
Convert decimal values \(0, 1, 2, 3, 4\) to Binary code (3 bits):
\(0 \to 000\)
\(1 \to 001\)
\(2 \to 010\)
\(3 \to 011\)
\(4 \to 100\)
Multi-bit difference example: Transition from \(3 \to 4\): \(011 \to 100\). Three bits change.
Gray Coding in PAM#
One of the most effective ways to assign bit sequences to amplitude levels is Gray coding.
In Gray coding for PAM, adjacent amplitude levels differ by only one binary digit.
Importance of Gray Coding in Demodulation#
During demodulation, received signals are often affected by noise, causing occasional misclassification of symbols. If Gray coding is used:
In the presence of noise, the most likely demodulation errors occur when the receiver selects an adjacent signal amplitude instead of the correct one.
Since adjacent amplitudes differ by only one bit, an erroneous decision results in only a single-bit error, reducing the impact of noise-induced errors.
In another word, in case of symbol error (selecting the wrong amplitude) results in a single-bit error.
In contrast, with binary coding, a symbol error may lead to multiple bit errors, leading to a higher bit error rate (BER) for the same symbol error rate (SER).
Signal Space of PAM#
One-Dimensional Vector Representation of PAM Signals
As established earlier, Pulse Amplitude Modulation (PAM) signals are one-dimensional, meaning that each signal can be represented as a single scalar value in signal space. The transmitted signals take the form:
when using bandpass pulse signal \(p(t)\), and:
when using lowpass pulse signal \(g(t)\).
Euclidean Distance in PAM#
The Euclidean distance between any two transmitted symbols \( s_m \) and \( s_n \) in the PAM constellation is given by:
Substituting the PAM signal representation:
for using bandpass pulse, and
for using baseband pulse.
This distance metric is critical in determining the error performance of the modulation scheme. The probability of symbol detection errors is heavily influenced by the spacing between constellation points: larger distances result in fewer errors, as the receiver can more reliably distinguish between different symbols.
Minimum Distance in the PAM Constellation#
For adjacent signal points in a PAM constellation, the amplitude levels are separated by:
where \( A_m, A_n \in \{\pm1, \pm3, \dots, \pm(M - 1)\} \).
The minimum Euclidean distance between adjacent signal points in the PAM constellation is determined by the pulse shape used:
For Bandpass Pulse \( p(t) \):
The minimum Euclidean distance is:
\[\boxed{ d_{\text{min}} = 2 \sqrt{\mathcal{E}_p} } \]Substituting \( \mathcal{E}_p = \frac{\mathcal{E}_g}{2} \), we get:
For Baseband Pulse \( g(t) \):
The energy of the baseband pulse is \( \mathcal{E}_g \), and the minimum Euclidean distance is directly:
\[\boxed{ d_{\text{min}} = \sqrt{2\mathcal{E}_g} } \]
Since \(\mathcal{E}_{\text{b,avg}} = \frac{(M^2 - 1)\mathcal{E}_g}{6 \log_2 M}\), we have
This expression shows how \( d_{\text{min}} \) depends on the constellation size \( M \), the average bit energy \( \mathcal{E}_{\text{b,avg}} \), and the logarithm of the number of symbols.
Significance
The minimum distance \( d_{\text{min}} \) is a critical parameter for determining the bit error rate (BER) in a noisy channel:
Larger \( d_{\text{min}} \): Implies better noise immunity, reducing the likelihood of symbol misclassification at the receiver.
Smaller \( d_{\text{min}} \): Increases sensitivity to noise, resulting in higher error rates.
Double-Sideband (DSB) and Single-Sideband (SSB) PAM Signals#
Double-Sideband (DSB) PAM Signal#
As we defined above, the equation for a Double-Sideband (DSB) PAM signal is given by:
where:
\( A_m \): Amplitude level of the signal.
\( g(t) \): Baseband pulse shape.
\( \cos(2\pi f_c t) \): Carrier signal modulating \( g(t) \).
\( f_c \): Carrier frequency.
Bandwidth of DSB PAM#
When \( g(t) \) is modulated onto the carrier \( \cos(2\pi f_c t) \), the spectrum of the resulting DSB signal contains:
A lower sideband centered at \( f_c - B \),
An upper sideband centered at \( f_c + B \), where \( B \) is the bandwidth of the baseband signal \( g(t) \).
Since the DSB signal includes both sidebands, it requires twice the bandwidth of the baseband signal \( g(t) \).
Single-Sideband (SSB) PAM Signal#
The SSB PAM signal is derived by suppressing one of the sidebands (either the upper or lower sideband) in the modulated signal. The equation for an SSB PAM signal is:
where:
\( g(t) \): Baseband pulse shape.
\( \hat{g}(t) \): Hilbert transform of \( g(t) \).
The \( \pm \) sign selects the upper sideband (USB) or lower sideband (LSB).
\( e^{j2\pi f_c t} \): Complex carrier modulation.
Bandwidth of SSB PAM#
The SSB signal retains only one of the two sidebands (either upper or lower).
Therefore, it requires only the bandwidth of the baseband signal \( g(t) \).
Here, we have:
Hilbert Transform:
The term \( \hat{g}(t) \) (Hilbert transform of \( g(t) \)) is used to construct a complex representation of the signal, allowing for single-sideband modulation.
Bandwidth Comparison:
DSB PAM: Uses both sidebands, requiring \( 2B \) bandwidth.
SSB PAM: Uses only one sideband, requiring \( B \) bandwidth.
Efficiency:
SSB PAM is more bandwidth-efficient than DSB PAM but may be more complex to implement due to the requirement of the Hilbert transform and precise sideband suppression.
The choice between DSB and SSB PAM depends on the trade-off between bandwidth efficiency and implementation complexity. DSB is simpler but uses twice the bandwidth, while SSB achieves better bandwidth utilization by halving the required bandwidth.